Time for Half of a Bacterial Population to Become Phage-Adsorbed
by Stephen T. Abedon Ph.D. (abedon.1@osu.edu)
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Calculates how long it takes for half of the bacterial population to be adsorbed, assuming phage titer remains constant throughout. This simpler closed-form model is appropriate when phage losses to adsorption are negligible (phages greatly outnumber bacteria), or when phages are continuously replenished — for example by in situ phage replication or repeated dosing that replaces virions lost to adsorption. The Phages Decline tab provides a more realistic numerical model for situations where phage depletion is non-negligible and replenishment does not occur.
Calculates the bacterial half-life numerically, accounting for phage depletion as phages adsorb bacteria. Both the free phage titer and the unadsorbed bacterial fraction decline simultaneously. This model is more appropriate when bacterial concentrations are not negligible relative to phage numbers and phages are not being replenished. See the Constant Phages tab for the simpler closed-form model.
When phages encounter bacteria, individual phage particles adsorb to — and thereby effectively commit to kill or at least commit to kill — individual bacterial cells. Assuming a constant phage titer P (PFU mL−1) and a given adsorption rate constant k (mL min−1), the rate of phage adsorption per bacterium follows first-order kinetics:
Because the fraction of bacteria not yet adsorbed declines exponentially with time, a bacterial half-life can be defined analogously to radioactive decay:
This concept — and the default parameter values used in this calculator — are discussed and illustrated in Abedon [1], which uses k = 2.5 × 10−9 mL min−1 (from Stent) and presents bacterial half-lives as a function of phage concentration.
The Constant Phages calculation assumes the phage titer P remains constant throughout the interval. This is a reasonable approximation when bacteria are greatly outnumbered by phages, so phage losses to adsorption are minor. It is also applicable when phages are being actively replenished — for example by repeated dosing, or by in situ phage replication (see Phage Therapy Relevance below). When this assumption does not hold, the Phages Decline tab provides a numerical model accounting for simultaneous depletion of both phage and bacterial populations.
When the bacterial concentration is not negligible relative to the phage titer, the phage population is depleted as adsorptions occur, causing the adsorption rate to slow over time. The Phages Decline tab models this mutual depletion numerically: at each time step, both the free phage titer and the unadsorbed bacterial fraction are updated based on the current adsorption rate. The result is always a longer half-life than the constant-phage approximation predicts, because the effective phage titer is declining throughout the interval. The degree of extension depends on the initial ratio of bacteria to phage — the higher the bacterial concentration relative to phages, the greater the extension.
In phage therapy, particularly inundative (non-replicative) treatment, it is important to deliver a phage dose sufficient to adsorb bacteria quickly enough to prevent proliferation and toxin production. The bacterial half-life provides an intuitive measure of how rapidly a given phage dose can engage the target population. A short half-life — achieved through a high phage titer or a large k — indicates rapid engagement.
It should be noted that in situ phage replication — as occurs during active treatment — can in principle replenish phage numbers lost to adsorption, so long as phage replication rates equal or exceed adsorption losses. While this balance cannot be maintained indefinitely, for as long as it holds the Constant Phages calculation may be more relevant than the Phages Decline model.
The adsorption rate constant k is a property of the specific phage–host pair under given environmental conditions (temperature, ionic strength, etc.). It can be calculated using the Adsorption Rate Calculator based on experimentally derived adsorption rates. Typical values range from approximately 10−10 to 10−8 mL min−1; values substantially exceeding 10−8 mL min−1 are unusual for phage adsorption. The default value used here, k = 2.5 × 10−9 mL min−1, reflects a commonly cited value from Stent, as used in Abedon [1].
The calculations here assume that all phages adsorb at the same rate, yielding simple exponential (monophasic) kinetics. In practice, adsorption curves are sometimes multiphasic — different subpopulations of phage particles adsorb at measurably different rates, due to heterogeneity in phage particle structure, partial inactivation, or variation in receptor availability. When adsorption is multiphasic, a single k value will not fully describe the kinetics, and half-life estimates from this calculator should be interpreted with appropriate caution. For discussion of multiphasic adsorption and methods for estimating adsorption rate constants, see the Adsorption Rate Calculator.
| Symbol | Meaning | Units |
|---|---|---|
| t0.5 bacteria | Bacterial half-life | min (consistent with k) |
| ln(2) | Natural log of 2 ≈ 0.6931 | dimensionless |
| k | Phage adsorption rate constant | mL min−1 |
| P | Phage titer (assumed constant) | PFU mL−1 |
Let B(t) be the fraction of bacteria not yet adsorbed at time t. With phage titer P held constant, each bacterium is adsorbed at rate k × P. The governing differential equation is:
Integrating with B(0) = 1:
Setting B(t) = 0.5 and solving for t:
When phage depletion is accounted for, both the free phage titer and the unadsorbed bacterial fraction decline simultaneously. Let P(t) be the free phage concentration and N(t) the unadsorbed bacterial concentration at time t, with initial values P0 and N0. Each adsorption event removes one free phage and commits one bacterium, giving the coupled system:
Both equations share the same right-hand side, so their difference is conserved:
Substituting P(t) = N(t) + Δ into the equation for N:
This separable ODE has an exact analytical solution when Δ ≠ 0. For generality (including the edge case P0 = N0, Δ = 0), this calculator solves it numerically using the Euler method with a small time step. The half-life is the first time at which the unadsorbed bacterial fraction falls to 0.5. Because the phage titer declines throughout, the result is always equal to or longer than the constant-phage estimate.
Rearranging the constant-phage formula gives the phage titer or k required to achieve a 10-minute half-life:
These are displayed in the Constant Phages tab after each calculation.